Defining a full independence axiom in the Savagean multi-profile framework

Determine whether a full independence axiom—requiring social rankings to be invariant when two arbitrary acts are mixed with a common proportion—can be appropriately defined within the multi-profile Savage framework used in this paper by constructing mixture operations between two arbitrary acts (not only between an act and a constant act) and specifying conditions under which such an axiom is well-defined and applicable to aggregation rules under uncertainty.

Background

The paper introduces pseudo-mixed acts to mimic mixture operations in Savage’s framework, enabling axioms such as restricted certainty independence and certainty independence that mix an act with a constant act. Proposition 1 guarantees the existence of pseudo-mixed acts only for combinations of an act and a constant act, not for arbitrary pairs of acts.

In standard Anscombe–Aumann frameworks, full independence concerns mixtures of arbitrary acts and is central to additive representations. Extending this to the current uncertain, multi-profile Savagean setting would require a general mixture operation for arbitrary acts. The authors note that establishing such an operation—and thereby formulating a full independence axiom—remains unresolved, which limits the scope of independence-based characterizations in their framework.